This book is intended to be the text for a “capstone” course, that is, the final course taken by undergraduate math majors, that should in some way tie together all the preceding courses. The book does this by explicitly drawing connections between branches of mathematics, and most chapters relate linear algebra to one other branch of mathematics. The book gives a competent exposition of a variety of subjects, is well-supplied with examples illustrating the definitions, and has a modest number of (rather easy) exercises.
My big gripe with this book is that, after developing relationships between different branches of mathematics, it does not do anything to use these new structures to solve problems in the original branches. (Chapter 1 is an exception: it shows how geometry can be tied to algebra to determine which angles can be trisected, a very old problem that was intractable until these connections were invented.) I’ll pick on Chapter 5 as an example. This chapter, “Matrices and Topology,” shows how to define a metric on spaces of matrices and examines the topological properties of these spaces, especially investigating their connected components. Then it stops. Do we know any more about matrices, or topology, when we are done? Not really. There are germs of linear operator theory and Lie groups here, subjects that are useful in studying integral and differential equations and many areas of mathematical physics. Making this connection has historically been a fruitful idea, but none of that is mentioned here.
I think much of the reason the book does not work on such specific problems is that it is focused on classification and categorization. The book’s attitude is captured on p. 192 when it says “The ideal goal of all areas of pure mathematics is to produce lists of properties that can be attached to the objects being studied such that these lists characterize when two objects are equivalent.” Many mathematicians would disagree that this is the only or main goal of pure mathematics. Certainly it doesn’t cover such triumphs as the Prime Number Theorem or the Fundamental Theorem of Algebra (to name two problems that were solved by connecting the original problem to other areas of mathematics).
The prose tends to be verbose (the statement I quoted above about the purpose of pure mathematics is typical). I noticed a half-dozen or so minor errors, mostly involving in leaving out parts of a hypothesis or getting things backward.
A very different text for a capstone course, that also emphasizes the unity of mathematics and drawing connections, is Iosevich’s A View from the Top: Analysis, Combinatorics, and Number Theory. Iosevich’s book takes the opposite approach from the present book: it starts with specific difficult problems and studies how to bring different branches of mathematics to bear on them, rather than bringing together the branches first. This is a much more realistic problem-solving approach and makes the subject much easier to motivate.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.